Twisted photonic Weyl meta-crystals and aperiodic Fermi arc scattering

As a milestone in the exploration of topological physics, Fermi arcs bridging Weyl points have been extensively studied. Weyl points, as are Fermi arcs, are believed to be only stable when preserving translation symmetry. However, no experimental observation of aperiodic Fermi arcs has been reported so far. Here, we continuously twist a bi-block Weyl meta-crystal and experimentally observe the twisted Fermi arc reconstruction. Although both the Weyl meta-crystals individually preserve translational symmetry, continuous twisting operation leads to the aperiodic hybridization and scattering of Fermi arcs on the interface, which is found to be determined by the singular total reflection around Weyl points. Our work unveils the aperiodic scattering of Fermi arcs and opens the door to continuously manipulating Fermi arcs.


I. Effective medium of Weyl metamaterials
We start from effective media theory, the constitutive relation is given as, where the sign convention is (, ) =  "  $(&'(•) .For the top Weyl metamaterial (fixed), we have, where for simplicity, we already assume  " =  " =  = 1 and  = 1,  = 0.9,  = 1,  " = 1.Components in the constitutive relation is given as The Weyl nodes derived from Eq. S1 consist of a transverse mode and a longitude mode 1 , where the crossing of two modes forms Weyl nodes.The nonlocal effect, resulting from inter-unit-cell interaction, bends the longitudinal mode to be dispersive with negative group velocity, and slightly moves the Weyl nodes.Due to the nonlocal effect,  is replaced by , where  = 1,  = 10,  = 1.Thus,  - and  -are given as , respectively.Then the twisted Fermi arcs (see Fig. 1b) from the effective medium perspective are shown in Supplementary Figure 1.
Supplementary Figure 1.Twisted Fermi arcs in the twisted bi-block Weyl metamaterials.

II. Reflection matrix
The reflection of the bottom meta-crystal is discussed first.We need two modes to form the basis for electromagnetic waves.Here we use (| Note that the state space of incident and reflected basis are different.We introduce a mirror operator  Z to connect them, The matrix representation of  Z is, For simplicity, we choose (| >?@3 ⟩, | >?@3 ⟩) as the basis for downward incidence, and take  g as the reflection operator, the reflection matrix is constructed by, Further applying mirror  Z to connect the two basis, the matrix representation of  g  Z . Then, we obtain the reflection matrix as, Note that the reflection matrix is the representation of  g  Z rather than  g .For a total reflection, the reflection matrix o g  Z p is a unitary matrix.
Next, we consider a wave propagating along a round-trip, i.e., experiencing twice reflections.Note that for the round-trip, we do not need to connect two basis state spaces, that is, the state space is always the same after the round-trip.We define  g ′ as the reflection operator of the top meta-crystal, Finally, we have the matrix representation by following  g The round-trip reflection matrix describes the two reflection processes, which can be calculated separately and combined simply using matrix multiplication.
effect, we assume that the entry  -is related to the in-plane momentum ( < ,  = ).Similarly expanding the electromagnetic field as,
The longitudinal field component is given as, Finally, the coupled equations are, where, Since the bottom Weyl meta-crystal is twisted by an angle , the constitutive matrices are, where, The coupled equation for the bottom meta-crystal could be constructed following the above process in a similar way, and  is the twisted angle.

Round-trip eigen phase
Here, in order to obtain the twisted Fermi arcs, we want the round-trip eigen phases of the spacer cavity between the two twisted Weyl meta-crystals.The top and bottom blocks are totally reflected, and a semi-infinite condition is used to ensure that the surface states decay upwards and downwards, respectively.The thickness of spacer layer is tunable and can be zero as a critical condition.

VIII. Fermi arcs arising from scattering
In effective media theory, there are only four Weyl points for each Weyl block as shown in Fig. 1a.Considering the weak periodic modulation, more Weyl points are introduced.
Specifically,  plane waves introduce to 4 Weyl points, that is, effective media theory counts only the zeroth plane wave.For small twisting angles , the Fermi arcs appearing in the top FBZ are dominated by the zeroth plane wave.With increasing the twisting angle  , Fermi arcs tend to bridge high-order Weyl points arising from scattering and reconstruct across the top FBZ, as shown in Supplementary Figures 8a   and b.
As indicated by the dashed arrow in Supplementary Figure 8a , we only introduce refractive index modulation on  . .RCWA method requires Fourier decomposition of the dielectric function  .= ∑ ̅ .() •  $• ∈P , and  is the in-plane coordinate in real space, and  ∈  are the reciprocal lattice vectors.

,
Fermi arc runs across the top FBZ, bridging another Weyl point outside.With increasing frequency, the Fermi arc rotates clockwise and gets close to another Fermi arc.The corresponding experimental mapping of Fermi arc evolution is another strong support of our RCWA method.In Supplementary Figure 8a, we manipulate the connectivity of Fermi arcs by shift the trivial phase  KK .As previously mentioned, the adjustment of the phase  KK , variation of the twist angle, and modification of frequencies are viable approaches for reconstructing Fermi arcs.